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A368608 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = k, where x,y,z are in {1,2,...,n} and x != y and y <= z.

%I A368608 #9 Jan 29 2024 11:01:49
%S A368608 2,1,4,5,2,1,6,9,8,4,2,1,8,13,14,12,6,4,2,1,10,17,20,20,16,9,6,4,2,1,
%T A368608 12,21,26,28,26,21,12,9,6,4,2,1,14,25,32,36,36,33,26,16,12,9,6,4,2,1,
%U A368608 16,29,38,44,46,45,40,32,20,16,12,9,6,4,2,1,18,33
%N A368608 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = k, where x,y,z are in {1,2,...,n} and x != y and y <= z.
%C A368608 Row n consists of 2n positive integers.
%e A368608 First six rows:
%e A368608    2   1
%e A368608    4   5   2   1
%e A368608    6   9   8   4   2   1
%e A368608    8  13  14  12   6   4   2  1
%e A368608   10  17  20  20  16   9   6  4  2  1
%e A368608   12  21  26  28  26  21  12  9  6  4  2  1
%e A368608 For n=2, there are 3 triples (x,y,z) having x != y and y <= z:
%e A368608   122:  |x-y| + |y-z| = 1
%e A368608   211:  |x-y| + |y-z| = 1
%e A368608   212:  |x-y| + |y-z| = 2
%e A368608 so row 2 of the array is (2,1), representing two 1s and one 2.
%t A368608 t1[n_] := t1[n] = Tuples[Range[n], 3];
%t A368608 t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] <= #[[3]] &];
%t A368608 a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t A368608 u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
%t A368608 v = Flatten[u]  (* sequence *)
%t A368608 Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]]  (* array *)
%Y A368608 Cf. A005443 (column 1), A027480 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368606, A368607, A368609.
%K A368608 nonn,tabf
%O A368608 1,1
%A A368608 _Clark Kimberling_, Jan 25 2024